Research Article
Refinements and Reverses of Tensorial and Hadamard Product Inequalities for Selfadjoint Operators in Hilbert Spaces Related to Young's Result
Abstract
Let $H$ be a Hilbert space. In this paper we show among others that, if the
selfadjoint operators $A$ and $B$ satisfy the condition $0$ $<$ $m\leq A,$ $B\leq
M,$ for some constants $m,$ $M,$ then
\begin{align*}
0& \leq \frac{m}{M^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}\otimes
1+1\otimes B^{2}}{2}-A\otimes B\right) \\
& \leq \left( 1-\nu \right) A\otimes 1+\nu 1\otimes B-A^{1-\nu }\otimes
B^{\nu } \\
& \leq \frac{M}{m^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}\otimes
1+1\otimes B^{2}}{2}-A\otimes B\right)
\end{align*}
for all $\nu \in \left[ 0,1\right] .$ We also have the inequalities for
Hadamard product
\begin{align*}
0& \leq \frac{m}{M^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}+B^{2}}{2}%
\circ 1-A\circ B\right) \\
& \leq \left[ \left( 1-\nu \right) A+\nu B\right] \circ 1-A^{1-\nu }\circ
B^{\nu } \\
& \leq \frac{M}{m^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}+B^{2}}{2}%
\circ 1-A\circ B\right)
\end{align*}
for all $\nu \in \left[ 0,1\right] .$
References
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- [2] M. Tominaga, Specht’s ratio in the Young inequality, Sci. Math. Japon., 55 (2002), 583–588.
- [3] S. Furuichi, Refined Young inequalities with Specht’s ratio, Journal of the Egyptian Mathematical Society, 20(2012), 46–49.
- [4] F. Kittaneh, Y. Manasrah, Improved Young and Heinz inequalities for matrix, J. Math. Anal. Appl., 361 (2010), 262–269.
- [5] F. Kittaneh, Y. Manasrah, Reverse Young and Heinz inequalities for matrices, Linear Multilinear Algebra., 59 (2011), 1031–1037.
- [6] G. Zuo, G. Shi, M. Fujii, Refined Young inequality with Kantorovich constant, J. Math. Inequal., 5 (2011), 551–556.
- [7] W. Liao, J. Wu, J. Zhao, New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant, Taiwanese J. Math., 19(2) (2015), 467–479.
- [8] S. S. Dragomir, A note on Young’s inequality, Revista de la Real Academia de Ciencias Exactas, F´ısicas y Naturales. Serie A. Matematicas, 111(2) (2017), 349–354. Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 126. [http://rgmia.org/papers/v18/v18a126.pdf].
- [9] S. S. Dragomir, A note on new refinements and reverses of Young’s inequality, Transyl. J. Math. Mec. 8(1) (2016), 45–49. Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. [https://rgmia.org/papers/v18/v18a131.pdf].
- [10] H. Alzer, C. M. da Fonseca, A. Kovacec, Young-type inequalities and their matrix analogues, Linear and Multilinear Algebra, 63(3) (2015), 622–635.
- [11] S. Furuichi, N. Minculete, Alternative reverse inequalities for Young’s inequality, J. Math Inequal., 5(4) (2011), 595–600.
- [12] H. Araki, F. Hansen, Jensen’s operator inequality for functions of several variables, Proc. Amer. Math. Soc., 128(7) (2000), 2075–2084.
- [13] A. Koranyi, On some classes of analytic functions of several variables, Trans. Amer. Math. Soc., 101 (1961), 520–554.
- [14] T. Furuta, J. Micic Hot, J. Pecaric, Y. Seo, Mond-Peˇcari´c Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
- [15] S. Wada, On some refinement of the Cauchy-Schwarz inequality, Lin. Alg. & Appl., 420 (2007), 433–440.
- [16] J. I. Fujii, The Marcus-Khan theorem for Hilbert space operators, Math. Jpn., 41(1995), 531–535.
- [17] T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Lin. Alg. & Appl., 26 (1979), 203–241.
- [18] J. S. Aujila, H. L. Vasudeva, Inequalities involving Hadamard product and operator means, Math. Japon., 42 (1995), 265–272.
- [19] K. Kitamura, Y. Seo, Operator inequalities on Hadamard product associated with Kadison’s Schwarz inequalities, Scient. Math., 1(2) (1998), 237–241.
Year 2024,
Volume: 7 Issue: 1, 56 - 70, 04.03.2024
Abstract
References
- [1] W. Specht, Zer Theorie der elementaren Mittel, Math. Z., 74 (1960), 91–98.
- [2] M. Tominaga, Specht’s ratio in the Young inequality, Sci. Math. Japon., 55 (2002), 583–588.
- [3] S. Furuichi, Refined Young inequalities with Specht’s ratio, Journal of the Egyptian Mathematical Society, 20(2012), 46–49.
- [4] F. Kittaneh, Y. Manasrah, Improved Young and Heinz inequalities for matrix, J. Math. Anal. Appl., 361 (2010), 262–269.
- [5] F. Kittaneh, Y. Manasrah, Reverse Young and Heinz inequalities for matrices, Linear Multilinear Algebra., 59 (2011), 1031–1037.
- [6] G. Zuo, G. Shi, M. Fujii, Refined Young inequality with Kantorovich constant, J. Math. Inequal., 5 (2011), 551–556.
- [7] W. Liao, J. Wu, J. Zhao, New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant, Taiwanese J. Math., 19(2) (2015), 467–479.
- [8] S. S. Dragomir, A note on Young’s inequality, Revista de la Real Academia de Ciencias Exactas, F´ısicas y Naturales. Serie A. Matematicas, 111(2) (2017), 349–354. Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 126. [http://rgmia.org/papers/v18/v18a126.pdf].
- [9] S. S. Dragomir, A note on new refinements and reverses of Young’s inequality, Transyl. J. Math. Mec. 8(1) (2016), 45–49. Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. [https://rgmia.org/papers/v18/v18a131.pdf].
- [10] H. Alzer, C. M. da Fonseca, A. Kovacec, Young-type inequalities and their matrix analogues, Linear and Multilinear Algebra, 63(3) (2015), 622–635.
- [11] S. Furuichi, N. Minculete, Alternative reverse inequalities for Young’s inequality, J. Math Inequal., 5(4) (2011), 595–600.
- [12] H. Araki, F. Hansen, Jensen’s operator inequality for functions of several variables, Proc. Amer. Math. Soc., 128(7) (2000), 2075–2084.
- [13] A. Koranyi, On some classes of analytic functions of several variables, Trans. Amer. Math. Soc., 101 (1961), 520–554.
- [14] T. Furuta, J. Micic Hot, J. Pecaric, Y. Seo, Mond-Peˇcari´c Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
- [15] S. Wada, On some refinement of the Cauchy-Schwarz inequality, Lin. Alg. & Appl., 420 (2007), 433–440.
- [16] J. I. Fujii, The Marcus-Khan theorem for Hilbert space operators, Math. Jpn., 41(1995), 531–535.
- [17] T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Lin. Alg. & Appl., 26 (1979), 203–241.
- [18] J. S. Aujila, H. L. Vasudeva, Inequalities involving Hadamard product and operator means, Math. Japon., 42 (1995), 265–272.
- [19] K. Kitamura, Y. Seo, Operator inequalities on Hadamard product associated with Kadison’s Schwarz inequalities, Scient. Math., 1(2) (1998), 237–241.
There are 19 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Pure Mathematics (Other) |
| Journal Section | Articles |
| Authors | |
| Early Pub Date | March 4, 2024 |
| Publication Date | March 4, 2024 |
| Submission Date | September 19, 2023 |
| Acceptance Date | February 19, 2024 |
| Published in Issue | Year 2024 Volume: 7 Issue: 1 |
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